Cost function deformation in quantum approximate optimization

ABSTRACT

Techniques for performing cost function deformation in quantum approximate optimization are provided. The techniques include mapping a cost function associated with a combinatorial optimization problem to an optimization problem over allowed quantum states. A quantum Hamiltonian is constructed for the cost function, and a set of trial states are generated by a physical time evolution of the quantum hardware interspersed with control pulses. Aspects include measuring a quantum cost function for the trial states, determining a trial state resulting in optimal values, and deforming a Hamiltonian to find an optimal state and using the optimal state as a next starting state for a next optimization on a deformed Hamiltonian until an optimizer is determined with respect to a desired Hamiltonian.

BACKGROUND

The subject disclosure relates to quantum computing, and morespecifically, to solving combinatorial optimization problems withquantum circuits.

SUMMARY

The following presents a summary to provide a basic understanding of oneor more embodiments of the invention. This summary is not intended toidentify key or critical elements, or delineate any scope of theparticular embodiments or any scope of the claims. Its sole purpose isto present concepts in a simplified form as a prelude to the moredetailed description that is presented later. In one or more embodimentsdescribed herein, devices, systems, computer-implemented methods,apparatus and/or computer program products facilitating automatingquantum circuit debugging are described.

According to an embodiment, a system can comprise a mapping componentthat maps a cost functions to a Hamiltonian based on one or moreconstraints to map the optimization problem into an optimization problemover allowed quantum states, a trial state and measurement componentthat generates trial states corresponding to the Hamiltonian by physicaltime evolution of quantum hardware interspersed with control pulses toentangle qubits of the quantum hardware, and that measures a quantumcost function for the trial states to determine a trial state thatresults in optimal values, and a deformation component that deforms aHamiltonian into a deformed Hamiltonian to find an optimal state, anduses the optimal state as a next starting state for a next optimizationon the deformed Hamiltonian until an optimizer is determined withrespect to a desired Hamiltonian.

According to another embodiment, a computer-implemented method isprovided. The computer-implemented method can comprise mapping a costfunction associated with the combinatorial optimization problem to anoptimization problem over allowed quantum states, comprisingconstructing a quantum Hamiltonian for the cost function, and generatinga set of trial states by a physical time evolution of the quantumhardware interspersed with control pulses. Furthermore, thecomputer-implemented method can comprise measuring a quantum costfunction for the trial states, determining a trial state resulting inoptimal values, and deforming a Hamiltonian to find an optimal state andusing the optimal state as a next starting state for a next optimizationon a deformed Hamiltonian until an optimizer is determined with respectto a desired Hamiltonian.

According to yet another embodiment, a computer program productfacilitating solving a combinatorial optimization problem can beprovided, the computer program product comprising a computer readablestorage medium having program instructions embodied therewith, theprogram instructions executable by a processor to cause the processor tomap a cost function associated with the combinatorial optimizationproblem to an optimization problem over allowed quantum states,comprising constructing a quantum Hamiltonian for the cost function,generate a set of trial states by a physical time evolution of thequantum hardware interspersed with control pulses, measure a quantumcost function for the trial states, determine a trial state resulting inoptimal values, and deform a Hamiltonian to find an optimal state andusing the optimal state as a next starting state for a next optimizationon a deformed Hamiltonian until an optimizer is determined with respectto a desired Hamiltonian.

In yet another embodiment, a computer-implemented method can beprovided, comprising obtaining a starting Hamiltonian having associatedstarting control parameters. Aspects may include using quantum hardwareto deform the starting Hamiltonian into a deformed Hamiltonianassociated with optimal control parameters for that deformedHamiltonian, using the quantum hardware to repeatedly deform thedeformed Hamiltonian with the associated optimal control parameters forthat deformed Hamiltonian into further deformed Hamiltonians and furtheroptimal control parameters associated therewith until a desiredHamiltonian is reached, and outputting information corresponding to thedesired Hamiltonian and the optimal control parameters associated withthe desired Hamiltonian.

In another embodiment, a computer program product can be provided. Thecomputer program product can obtain a starting Hamiltonian havingassociated starting control parameters. The computer program product canuse quantum hardware to deform the starting Hamiltonian into a deformedHamiltonian associated with optimal control parameters for that deformedHamiltonian, use the quantum hardware to repeatedly deform the deformedHamiltonian with the associated optimal control parameters for thatdeformed Hamiltonian into further deformed Hamiltonians and furtheroptimal control parameters associated therewith until a desiredHamiltonian is reached, and output information corresponding to thedesired Hamiltonian and the optimal control parameters associated withthe desired Hamiltonian.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an example, non-limiting system thatillustrates various aspects of the technology in accordance with one ormore embodiments described herein.

FIG. 2 illustrates a diagram representing example componentscorresponding to the various technical aspects of FIG. 1 in accordancewith one or more embodiments described herein.

FIG. 3 illustrates an example diagram representing mapping of a costfunction into a Hamiltonian cost function in accordance with one or moreembodiments described herein.

FIG. 4 illustrates an example diagram representing aspects related toenergy measurement in accordance with one or more embodiments describedherein.

FIG. 5 illustrates an example diagram representing trial state-basedmeasurements using entanglement via quantum circuits in accordance withone or more embodiments described herein.

FIG. 6 illustrates an example diagram representing aspects related tooutputting an energy value based on energy measurement in accordancewith one or more embodiments described herein.

FIG. 7 illustrates an example diagram representing how classicaloptimization can be used as part of quantum approximate optimization inaccordance with one or more embodiments described herein.

FIG. 8 illustrates an example diagram representing cost functiondeformation to produce final states representing a cost functionsolution in accordance with one or more embodiments described herein.

FIGS. 9-11 illustrates example graphs showing optimized measurementprobabilities obtained by gradually deforming a function correspondingto a combinatorial optimization problem (MaxCut) for vertex sets of two,three and four vertices, respectively in accordance with one or moreembodiments described herein.

FIG. 12 illustrates a block diagram of an example, non-limiting systemthat facilitates optimization of an optimization problem in accordancewith one or more embodiments described herein

FIG. 13 illustrates a flow diagram of an example, non-limitingcomputer-implemented method facilitating optimization of an optimizationproblem in accordance with one or more embodiments described herein.

FIG. 14 illustrates a flow diagram of an example, non-limitingcomputer-implemented method facilitating optimization of an optimizationproblem via cost function deformation in accordance with one or moreembodiments described herein.

FIG. 15 illustrates a block diagram of an example, non-limitingoperating environment in which one or more embodiments described hereincan be facilitated.

DETAILED DESCRIPTION

The following detailed description is merely illustrative and is notintended to limit embodiments and/or application or uses of embodiments.Furthermore, there is no intention to be bound by any expressed orimplied information presented in the preceding Background or Summarysections, or in the Detailed Description section.

One or more embodiments are now described with reference to thedrawings, wherein like referenced numerals are used to refer to likeelements throughout. In the following description, for purposes ofexplanation, numerous specific details are set forth in order to providea more thorough understanding of the one or more embodiments. It isevident, however, in various cases, that the one or more embodiments canbe practiced without these specific details.

FIG. 1 illustrates an example, general framework directed towardssolving classical optimization problems, including by mapping anoptimization problem cost function 102 via a mapping component 104 to aHamiltonian cost function 106, as further described herein withreference to FIG. 3. Note that to represent the classical cost functionas a quantum Hamiltonian, the Hamiltonian can be expressed as a linearcombination of Pauli Z terms.

With a classical optimization problem/cost function mapped to aHamiltonian cost function, trial state and measurement operations (asdescribed with reference to FIGS. 4-6) can be performed to generate ashort depth quantum circuit that depends on a set of continuousparameters. To this end, a trial state and measurement component 107comprising the quantum computer/circuitry 108 can be used, for example,to sample from an optimal state in the computational basis to obtain bitstrings that provide a good approximation to the binary optimizationproblem and to measure the quantum cost function for trial states inorder to feed it to a classical optimization routine (as described ingreater detail with reference to FIG. 7).

In general, quantum computing employs quantum physics to encodeinformation, in contrast to binary digital techniques based ontransistors. For example, a quantum computer can employ quantum bits(e.g., qubits), which are basic units of quantum information. Qubitsoperate according to a superposition principle of quantum physics and anentanglement principle of quantum physics. The superposition principleof quantum physics states that each qubit can represent both a value of“1” and a value of “0” at the same time. The entanglement principle ofquantum physics states that qubits in a superposition can be correlatedwith each other. For instance, a state of a first value (e.g., a valueof “1” or a value of “0”) can depend on a state of a second value. Assuch, a quantum computer can employ qubits to encode information.

Aspects include modeling by microwave pulses sent to a superconductingquantum circuit that prepares a given quantum state(s) which are used todraw samples in optimization problems. As described with reference toFIGS. 4-6, a general approach is to take an initial guess of the set ofcontinuous parameters that generates a particular state, write theHamiltonian as a sum of energy measurements to obtain a state andaverage (e.g., sample from the state to measure average the Paulis), andreport back the average value.

The classical optimization component (block 114 in FIG. 1) of FIG. 7 canprovide a loop operation that starts with an initial guess of a set ofcontrol parameters, and provides the Hamiltonian along with the set ofparameters. A measurement component 110 obtains the energy value atblock 112 via a process described in more detail with reference to FIG.6. This energy value can be mapped to a new proposed update set of thecontrol parameters, and the loop repeats. Once the classicaloptimization converges (e.g., the energy value is not significantlychanging), the energy value is output in association with itscorresponding optimal control parameters. This optimization, forexample, minimizes locally for one given Hamiltonian and finds thecorresponding set of control parameters.

Block 118, described in greater detail with reference to FIG. 8comprises a deformation component directed to the deformation of thecost function as described herein. In general, for a Hamiltonian family116, the operations start with a cost function for which the exactcontrol parameters are known, and loops (by applying a circuit ofrepeated drift steps) to steadily deform that cost function via thequantum circuits to solve a more specific optimization problem. Ingeneral and as will be understood, aspects of the technology start bydeforming control parameters and tracking the cost function value untilreaching the final state 120. More particularly, when a Hamiltonianrepresenting an optimization problem is to be solved, a family ofHamiltonians (block 116) is chosen, starting with a simple Hamiltonianfor which the answer is known, which provides for correctly choosing thecontrol parameters knowing that the initial guess is very exact. Fromthat starting Hamiltonian, over time the Hamiltonian is graduallydeformed by permitted drift times to generate actual entanglement (FIG.5) until the Hamiltonian for which the minimum is being sought isreached, and then used to solve a classical optimization problem.

FIG. 2 shows an example of a processor 200 and memory 202 coupled tocomponents corresponding to some of the blocks of FIG. 1. As can beseen, a mapping component 204, trial state and measurement component207, classical optimization component 214 and cost function deformationcomponent 218 can perform various operations as described herein. As isunderstood, however, these are only examples, and at least some of theexemplified components can be combined into a lesser number ofcomponents and/or the exemplified components can be further separatedinto additional components, additional components may be present inalternative embodiments, and so on.

Turning to various examples and related details, in general,optimization or combinatorial optimization refers to searching for anoptimal solution in a finite or countably infinite set of potentialsolutions. Optimality is defined with respect to some criterion functionthat is to be minimized or maximized, which is typically called the costfunction. There are various types of optimization problems. Theseinclude minimization: cost, distance, length of a traversal, weight,processing time, material, energy consumption, number of objects, etc.;and maximization: profit, value, output, return, yield, utility,efficiency, capacity, number of objects, etc.

Any maximization problem can be cast in terms of a minimization problemand vice versa. Hence the most general form of a combinatorialoptimization problem is given by

Minimize C(x)

subject to x∈S

where x∈S, is a discrete variable and C:D→

is the cost function that maps from some domain D in to the real numbers

. Typically the variable x is subject to a set of constraints and lieswithin the set S of feasible points.

In binary combinatorial optimization problems, the cost function cantypically be expressed as a sum of terms that only involve subsets Q⊆[n]of the n bits in the string ∈ {0,1}^(n). The cost function C istypically written in the canonical form

${C(x)} = {\sum\limits_{Q}{w_{Q}\underset{i \in Q}{\Pi}x_{i}}}$

for x_(i)∈{0,1} and w_(Q)∈

. Optimization is directed towards finding the n-bit string x for whichC(x) is smallest.

FIG. 3 illustrates a block diagram of an example, non-limiting systemthat defines a way to represent a classical, combinatorial optimizationproblem on a quantum computer. In general, given a general cost function302 (e.g., a binary combinatorial optimization problem seeking binarystrings) that maps from some domain into real numbers, it is understoodthat many such cost functions typically have additional constraints 304.Those constraints can be constructed with helper functions, which ingeneral modify (block 306) the general cost function with penalty termsfor other helper functions. As described herein, the modified costfunction gets mapped to a diagonal Hamiltonian cost function 308 (adiagonal matrix), which sets the entries of the diagonal based on theoriginal problem, which can be represented with Pauli Z-terms as alsodescribed herein.

The cost function is thus mapped to a quantum problem by constructingfor every C a Hamiltonian diagonal in the computational basis. Theproblem Hamiltonian H_(C) associated to the cost function C(x) isdefined as

H _(C)=Σ_(x) C(x)|x

x|,

where x∈{0,1}^(n) labels the computational basis states |x

∈C^(d) where d=2^(n). The task of quantum optimization is then to find astate |ψ

∈Ω⊂C^(d) from a variational subclass Ω of quantum states that minimizesthe energy of H_(C).

Minimize

ψ|H _(C)|ψ

subject to |ψ

∈Ω.

As is seen via FIG. 3, the binary optimization problem has been turnedin to an optimization problem over allowed quantum states depending on atractable set of parameters that specify Ω. The state that minimizes theequation is denoted by |ω*

. Additional details are described below with reference to equations Eq.1-Eq. 4 (and the accompanying text) set forth herein.

Turning to FIGS. 4-6, with the classical optimization problem mapped toa Hamiltonian optimization problem, short depth circuits that depend ona set of continuous parameters can be generated. In general and asdescribed herein, this is accomplished by taking an initial guess of aset of continuous parameters (block 402) that generates a particularstate (block 404), and writing that Hamiltonian as a sum of Pauli-Zterms. In general, this is accomplished by obtaining state informationand averaging, that is, sample from that state/average those Paulis(blocks 406 and 408, looping back to block 404 for some number ofiterations), or in other words, measure those Paulis to obtain anaverage value and report that average measurement back (block 410).

Turning to trial states with reference to FIG. 5, unitary stateentanglement (U_(ent)) can be represented on a quantum computer, such asrepresented in the example short depth circuit 500. The short depthcircuit can be represented on a superconducting quantum computer, suchas with entangling gates 502 and 504. These entangling gates arecross-resonance gates that talk to multiple qubits, and may be subjectto certain parameters, such as a depth parameter (how many crossresonance gate interactions are wanted) and so forth. Note that suchclassical parameters thus specify a family of states U_(ent); furtherdetails are provided with reference to equations 8, 9 and 10 and theiraccompanying text. Unitary entanglement U_(ent) thus comes fromcross-resonance gates (generally defined in equations 10, 11 and 12) anda superconducting quantum computing chip. It should be noted that thisis only a non-limiting example; other gates that entangle can bealternatively used.

Returning to FIG. 4, block 408 that measures the energy is generallydescribed the flow diagram for measuring the energy of FIG. 6. Asdescribed herein, at block 602 a Hamiltonian is given as combination ofPauli terms (P_(α)'s) and real numbers (h_(α)'s). For each of the alphas(α's) (blocks 604 and 608), expectation values of the individual Paulisare measured (block 606) and then added up (block 610) to obtain a finalenergy (block 612).

As can be seen, there is a set of Pauli operators paired withcoefficients. A parameter is chosen and the Pauli-operator outcome ismeasured on individual quantum states, multiple times until the averageis obtained, with the h_(α)'s weights (the real number in front of theexpectation value) summed to total energy. The output is the sum ofthose individual measurements. Additional details are set forth withreference to equations 15 and 16 and their accompanying text.

Once provided with an energy value that depends on the initialparameters fed into a quantum device, a full minimization scheme may beperformed. To summarize, consider a set of trial states |ψ

that are generated by the physical time evolution of the quantumhardware interspersed with control pulses. These trial states Ω can bereferred to as hardware efficient trial states. The controlled physicaltime evolution is universal in that any state in C^(d) in principle canbe generated. The quantum computer is used to sample from the optimalstate |ψ*

in the computational basis |x

to obtain bit strings x that provide a good approximation to the binaryoptimization problem, and to measure the quantum cost function E=

ψ H_(C)ψ

for the trial states |ψ

∈Ω in order to feed it to a classical optimization routine.

FIG. 7 illustrates one such classical optimization loop, starting withan initial guess of parameters, to provide the Hamiltonian (block 702)with the set of control parameters (block 704). Block 706 provides aninitial guess of control parameters. Blocks 704, 708, 710 and 712describe a measurement loop to get the energy value out as in FIG. 4; inthe loop, this energy value gets mapped to a new proposed update set ofthe control parameters, and so on until convergence to a desired levelis reached. Note that the new set of control parameters depends on whichclassical optimization routine is being used, e.g., simulated annealing,stochastic gradient or the like, for continuous variables.

In general block 710 checks for convergence in that the energy does notsignificantly change (although the experiment may continue to run). Atconvergence, block 714 output the energy value along with itsassociated, corresponding optimal control parameters. This minimizeslocally for one given Hamiltonian and finds the control parameters basedon an initial guess of optimal control parameters.

Turning to cost function deformation as generally represented in FIG. 8,it is noted that although this quantum optimization problem is dependentin part on a classical optimizer, the optimization problem is differentfrom the original classical problem C because it is only over a class ofquantum states with (possibly) fewer parameters. FIG. 8 starts (block802) with choosing a family of Hamiltonians, including one in which theanswer is known. That is, the operations start with a simple (trivial)Hamiltonian and correctly chosen control parameters such that theinitial guess is very exact. From that trivial Hamiltonian and thecorrect initial guess, the operations gradually deform the Hamiltonianuntil reaching the Hamiltonian for which the minimum is being sought,corresponding to the classical optimization problem cost function to besolved (by sampling from the quantum optimization problem).

FIG. 8 minimizes the family of Hamiltonians (similar to FIG. 7) from s=0to 1 as increased by ε) by entering (block 804) with new controlparameters found for the previous s value at (s−ε) and deforming thevalue found (until s=1, block 806). In other words, the controlparameters are entered and minimized (e.g., via the loop in FIG. 7),entering initially with φ₀, θ₀ and the deformed Hamiltonian H performingthis local optimization loop (blocks 804, 806, 808 and 810).

Thus, described herein is deforming a family of Hamiltonians startingfrom a trivial Hamiltonian for which the ground state is known, andtracking the ground state as deformation occurs until a desired point isreached. By starting with a set of known parameters, and deforming theHamiltonian that depends on these parameters to find the next set ofoptimal parameters, the next set can be fed into a next optimizationloop for the next deformed Hamiltonian until reaching the desiredHamiltonian. At that point, the final optimal parameters for this finalHamiltonian are known, and can be used to sample to obtain the correctbit string.

Thus, a family of cost functions E(s)=

ψ|H(s)∥ψ

for s∈[0,1] can be deformed from a trivial trial state for which theoptimal state |ψ_(s=0)*

∈Ω and E(0)=

ψ*₀|H(0)|ψ*₀

is known. As an example, consider the family of Hamiltonians

${H(s)} = {{sH}_{C} - {( {1 - s} ){\sum\limits_{i = 1}^{n}\; X_{i}}}}$

for n bits, and the Pauli spin operator

$X = \begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}$

acting on every single site i. Described herein is minimizing the costfunction for parameters s, starting at s₀=0 and then using the optimal|ψ_(s)*

as the next starting state for the next optimization at s_(k+1)=s_(k)+εuntil s=1 is reached and the optimizer of E(1)=

ψ|H_(C)|ψ

is found. At every step the quantum hardware is used to evaluate thereal valued cost function E(s)=

ψ|H(s)|ψ

.

Note that access to the quantum system used for the optimization, andaccess to the method for optimizing the quantum cost function may beprovided through the internet. For example, and an interface may beprovided so that this routine can be accessed in the cloud.

A numerical investigation has been performed for small system sizes anda depth d=2 circuit with two entangling steps U_(ent) and and two setsof single qubit rotations U_(loc) ^(n) for the Hamiltonian family in Eq.(18). For a graph with two vertices and the corresponding Hamiltonian ofN=2 qubits, the optimized measurement probabilities

z₁, . . . , z_(N)|ψ({right arrow over (θ)}*_(s),{right arrow over(φ)}*_(s))

, are depicted in FIG. 9 for s∈[0,1]. The example of a three vertexgraph is shown in FIG. 10, and for a four vertex graph in FIG. 11. Thelocal parameters were optimized using a classical simulated annealingtechnique.

Note that FIGS. 9-11 are for a system from two to four cubits oninstances of a combinatorial optimization problem MaxCut, describedbelow. In general, the graphs represent encoding in a particular bitstring, starting with initial superposition—the distribution from whichto sample minimizing bit strings, gradually deforming the function untilit starts concentrating with the correct bit-string in the end. This isexemplified by the early measurements providing mostly uniform results,but as the cost function deforms towards the desired one, theconcentration of weights can be seen.

Turning to additional details including equations, in binarycombinatorial optimization problems, the cost function C:{0,1}^(n)→

is typically written in the canonical form

$\begin{matrix}{{C(x)} = {\sum\limits_{Q \Subset {\lbrack n\rbrack}}{w_{Q}\underset{i \in Q}{\Pi}x_{i}}}} & (1)\end{matrix}$

for x_(i)∈{0,1} and w_(Q)∈

where typically the size of set Q is small relative to n. A task is tofind the n-bit string x∈{0,1}^(n) for which C(x) is extremal. A problemmay seek to optimize C with additional constraints on the bit string. Togive an example, consider the “Traveling Salesman Problem” given below.Additional constraints on the variable x can be stated in terms of aconstraint function

g _(i): {0,1}^(n)→

, for 1 . . . k.  (2)

for which

g _(i)(x)=0, for=1 . . . k.  (3)

to enforce the constraint. It is well known that such constraints can beenforced in terms of a modified cost function C_(A)(x)=C(x)+Σ_(i=1)^(k)A_(i)g_(i)(x), where the A_(i)∈

⁺ are chosen so that A_(i)>>ΔC and ΔC indicates the smallest differencebetween the discreet values of C.

This classical problem can be cast in terms of a quantum HamiltonianH_(C) associated to the cost function C_(λ)(x), that is diagonal in thecomputational basis |x

.

$\begin{matrix}{H_{C} = {\sum\limits_{x \in {\{{0,1}\}}^{n}}{{C_{\lambda}(x)}{x\rangle}{{\langle x}.}}}} & (4)\end{matrix}$

To represent the optimization as a quantum Hamiltonian it is convenientto express the optimization as a linear combination of Pauli Z terms.This can be achieved immediately when C_(λ)(x) is a polynomial of lowdegree polynomial in x. If this is the case, a simple substitution forevery binary variable as x_(i)=2⁻¹(1−Z_(i)) is sufficient. This mappingis illustrated in FIG. 3 as set forth herein.

Example A: Weighted MaxCut

MaxCut is an NP-complete problem, with applications in clustering,network science, and statistical physics. The formal definition of thisproblem is the following:

Consider a n-node non-directed graph G=(V,E) where |V|=n with edgeweights w_(ij)>0 for (i,j)∈E. A cut is defined as a partition of theoriginal set V into two subsets. The cost function to be optimized is inthis case the sum of weights of edges connecting points in the twodifferent subsets. For x_(i)∈{0,1} or x_(i)=1 to each node i one triesto maximize the global cost function

$\begin{matrix}{{{C(x)} = {\sum\limits_{{({i,j})} \in E}{w_{ij}{x_{i}( {1 - x_{j}} )}}}},} & (5)\end{matrix}$

with w_(ij) real. To provide a mapping to an Ising Hamiltonian, this canbe done with the assignment x_(i)=2⁻¹(1−Z_(i)). The Hamiltonian costHamiltonian H_(c) is then given by

$\begin{matrix}{{C(Z)} = {{{\sum\limits_{i < j}{\frac{w_{ij}}{2}( {1 - Z_{i}} )( {1 + Z_{j}} )}} + {\sum\limits_{1}{{w_{i}( {1 - Z_{i}} )}\text{/}2}}} = {{{- \frac{1}{2}}( {{\sum\limits_{i < j}{w_{ij}Z_{i}Z_{j}}} + {\sum\limits_{i}{w_{i}Z_{i}}}} )} + {const}}}} & (6)\end{matrix}$

where const=Σ_(i<j) w_(ij)/2+Σ_(i)w_(i)/2. Because this constant is notrelevant, the weighted MaxCut problem is equivalent to minimizing theIsing Hamiltonian

$\begin{matrix}{H = {{\sum\limits_{i}{w_{i}Z_{i}}} + {\sum\limits_{i < j}{w_{ij}Z_{i}{Z_{j}.}}}}} & (7)\end{matrix}$

Example B: Traveling Salesman Problem

To illustrate the mapping of optimization problems with constraints,discussed herein is the Traveling Salesman Problem (TSP). The TSP on thenodes of a graph asks to find the shortest Hamiltonian cycle in a graphG=(V,E) with n=|V| nodes and distances, w_(ij) (distance from vertex ito vertex j). A Hamiltonian cycle is described by N² variables x_(i,p),where i represents the node and p represents its order in a prospectivecycle. The decision variable takes the value 1 if the solution occurs atnode i at time order p. Every node can only appear once in the cycle andfor each time a node has to occur. This amounts to the two constraints

${\sum\limits_{i}x_{i,p}} = {1\mspace{14mu} {\forall p}}$${\sum\limits_{p}x_{i,p}} = {1\mspace{14mu} {\forall i}}$

For nodes in the prospective ordering, if x_(i,p) and x_(j,p+1) are both1, then there should be an energy penalty if (i,j)∉E (not connected inthe graph). The form of this penalty is

${{\sum\limits_{{({i,j})} \notin E}{\sum\limits_{p}{x_{i,p}x_{j,{p + 1}}}}} > 0},$

where it is assumed the boundary condition of the Hamiltonian cycle,(p=N)≡(p=0). However, here it will be assumed a fully connected graphthat does not include this term. The distance that needs to be minimizedis

${C(x)} = {\sum\limits_{({i,j})}{w_{ij}{\sum\limits_{p}{x_{i,p}{x_{j,{p + 1}}.}}}}}$

To account for these additional constraints the cost function ismodified and put together in a single objective function to be minimized

${{C(x)} = {{\sum\limits_{i,j}{w_{ij}{\sum\limits_{p}{x_{i,p}x_{j,{p + 1}}}}}} + {A{\sum\limits_{p}( {1 - {\sum\limits_{i}x_{i,p}}} )^{2}}} + {A{\sum\limits_{i}( {1 - {\sum\limits_{p}x_{i,p}}} )^{2}}}}},$

where A is a free parameter. The parameter A needs to be large enough sothat these constraints are respected. One way to do this is to choose Asuch that A>max(w_(ij)). Furthermore, since the problem allows thesalesperson to return to the original city, without loss of generality,it is possible to set x₀₀=1, x_(i0)=0∀i≠0, and x_(0p)=0∀p≠0. By doingthis, the objective functions becomes

${C(x)} = {{\sum\limits_{i,{j = 1}}^{N - 1}\; {w_{ij}{\sum\limits_{p = 1}^{N - 1}\; {x_{i,p}x_{j,{p + 1}}}}}} + {\sum\limits_{j = 1}^{N - 1}\; {w_{0j}x_{j,1}{\sum\limits_{i = 1}^{N - 1}\; {w_{i\; 0}x_{i,{N - 1}}}}}} + {A{\sum\limits_{p = 1}^{N - 1}\; ( {1 - {\sum\limits_{i = 1}^{N - 1}\; x_{i,p}}} )^{2}}} + {A{\sum\limits_{i = 1}^{N - 1}\; {( {1 - {\sum\limits_{p = 1}^{N - 1}\; x_{i,p}}} )^{2}.}}}}$

The unconstrained problem can be mapped to a quantum Hamiltonian withthe substitution x_(i)=2⁻¹(1−Z_(i)), and the solution will be found byminimizing an Ising-type Hamiltonian.

Thus, the problems considered herein can be mapped to a problemHamiltonian H. It is a goal to either determine a good approximation tothe value of the ground state energy E_(min) or sample from the shortdepth approximation |ψ_(min)

of the ground state of the Hamiltonian. An exact estimation of E_(min)or an exact preparation of |ψ_(min)

will in general not be possible based one complexity theoreticarguments. Hence the focus herein is on algorithms that prepareapproximations to both.

A coherently controllable quantum mechanical system is used, such as forexample a superconducting chip with N qubits to prepare a quantum state

|ψ({right arrow over (θ)},{right arrow over (φ)}

=U _(ent) ^((d))(φ_(d))U _(loc) ^((d))(θ_(d)) . . . U _(ent) ⁽¹⁾(φ₁)U_(loc) ⁽¹⁾(θ₁)|0^(N)

,  (8)

by applying a circuit of d repeated drift steps as depicted in FIG. 5that is comprised of local single qubit rotations

U _(loc) ^((t))(θ_(t))=⊗_(i=n) ^(n) U(θ_(i,t)) and U(θ_(i,t)) inSU(2),   (9)

parameterized by {right arrow over (θ)}∈

^(m).

Furthermore entangling interactions can be applied

$\begin{matrix}{{{U_{ent}^{(t)}( \phi_{t} )} = {\exp ( {{iK}( \phi_{t} )} )}},{{{where}\mspace{14mu} {K( \phi_{t} )}} = {\sum\limits_{\alpha}{{J_{\alpha}( \phi_{t} )}{\sigma (\alpha)}}}},} & (10)\end{matrix}$

that depend on some real parameters {right arrow over (φ)}∈

^(m) for every σ(α)∈

_(N). There exist multiple choices for the entangling unitaries U_(ent)^((t))(φ_(t)); the trial states considered can be seen as ageneralization of the known Quantum-Approximate-Optimization-Algorithmstates, where U_(ent) ^((t))(φ_(t))=exp(iφ_(t)H_(C)) and additionalrestrictions are placed on U_(loc) ^((t))(θ_(t)), such thatU(θ_(i,t))=exp(iθ_(t)x_(i)). Note that these restrictions can be relaxedto consider more general Ising type coupling graphs for U_(ent)^((t))(φ_(t)) that do not necessarily need to correspond to the problemHamiltonian H_(C).

Note that these interactions are not native to the superconductingcircuit considered. As an example to illustrate the method, and not byway of limitation, consider applying cross-resonance gates, whichimplies that the entangling Unitary between qubit k and l is generatedby the two-local effective Hamiltonian is given by

h _(k.l) =aZ _(k) +bZ ₁ +c _(k.l) Z _(k) Z _(l) +d _(k,l)(1−e _(k,l))X_(k) +d _(k,l) e _(k,l) X _(k) Z _(l).  (11)

The parameters a, b, c_(k.l),d_(k,l), e_(k,l) are determined by theactual hardware of the superconducting circuit. This means a naturaldrift evolution K, for example, is given by

$\begin{matrix}{{{K( \theta_{t} )} = {\sum\limits_{k < l}{{J_{k,l}( \theta_{t} )}h_{k,l}}}},} & (12)\end{matrix}$

where the J_(k,l)(θ_(t)) can be tuned by experiment.

This “bang bang” controlled state corresponds to a most general statethat can be prepared on the physical quantum hardware if limited to amaximal set of K subsequent iterations of applying control pulses andpermitted drift times to generate entanglement (FIG. 5). For the generalapproach to the known Quantum-Approximate-Optimization-Algorithm theinteraction Z_(l)Z_(m) has to be effectively generated from h_(k,l), byapplying bang bang control pulses, which extends the circuit depth andallows for the introduction of additional coherent errors by restrictingthe variation over the control parameters. In contrast, described hereinis optimizing the control pulses in the presence of the native driftHamiltonian directly. Because it is known that the set of drifts{h_(kl)} together with the single control pulses U_(loc)(θ) areuniversal, any state can be prepared this way with sufficient circuitdepth.

As described herein, optimization problems can be mapped to quantumIsing-type models. In order to perform an adiabatic deformation of thecost function as described herein, more general Hamiltonians need to beconsidered. That is, ground states of Hamiltonians that are notnecessarily diagonal in the Z-basis need to be constructed. In general,assume that the Hamiltonian can be decomposed into a sum of a fewmulti-qubit Pauli operators,

$\begin{matrix}{{H = {{\sum\limits_{\alpha}{h_{\alpha}{\sigma (\alpha)}\mspace{14mu} {with}\mspace{14mu} h_{\alpha}}}\mspace{14mu} \in {\mathbb{R}}}},{{\sigma (\alpha)} \in {_{N}.}}} & (13)\end{matrix}$

By way of example for illustration purposes and not by way oflimitation, consider the Pauli decomposition for the transverse Isingmodel with arbitrary two body spin couplings

$\begin{matrix}{{H(s)} = {{( {1 - s} ){\sum\limits_{k =}^{n}\; X_{k}}} + {\sum\limits_{i < j}J_{ij}Z_{i}{Z_{j}.}}}} & (14)\end{matrix}$

Note that this is not limited by the connectivity of the Hamiltonian Hand can consider completely non-local Pauli operators P_(α), because itis not planned to implement the Hamiltonian directly in hardware, butrather to measure the individual Pauli operators on a set ofexperimentally controlled trial states.

The energy of H can then be measured directly in by first preparing|ψ({right arrow over (θ)},{right arrow over (φ)})

and then sampling the expectation values for the individual σ(α) for allthe α's. The expectation value is then computed as

E({right arrow over (θ)},{right arrow over (φ)})=

ψ({right arrow over (θ)},{right arrow over (φ)})|H|ψ({right arrow over(θ)},{right arrow over (φ)})

.  (15)

The expectation value of the Hamiltonian from the individualPauli-measurements

ψ({right arrow over (θ)},{right arrow over (φ)})|σ(α)|ψ({right arrowover (θ)},{right arrow over (φ)})

is evaluated as

$\begin{matrix}{{\langle{{\psi ( {\overset{arrow}{\theta},\overset{arrow}{\phi}} )}{H}{\psi ( {\overset{arrow}{\theta},\overset{arrow}{\phi}} )}}\rangle} = {\sum\limits_{\alpha}{h_{\alpha}{{\langle{{\psi ( {\overset{arrow}{\theta},\overset{arrow}{\phi}} )}{{\sigma (\alpha)}}{\psi ( {\overset{arrow}{\theta},\overset{arrow}{\phi}} )}}\rangle}.}}}} & (16)\end{matrix}$

Consider M independent and identically distributed samples of themeasurement by a repeated prepare and measurement set up (FIG. 4), wherethe variational trial state in Eq. 8 is prepared M times and thePauli-operator measured directly. To obtain a bound on the number M ofsamples, consider asymptotic statistics and estimate the confidenceinterval by the variance. This means that any measurement that is toreproduce the expectation value E({right arrow over (θ)},{right arrowover (φ)}) up to error E should prepare at least M≥

(ε⁻²−2Var_(E)) samples, where Var_(E) indicates the variance of theenergy of the measurement scheme.

A classical optimization routine is needed that will converge to theoptimal parameter values ({right arrow over (θ)}*,{right arrow over(φ)}*). Several options exist that can be applied. By way of example andnot limitation, one approach is to perform simulated annealing for thecost function E({right arrow over (θ)},{right arrow over (φ)}). Otheralternatives include the SPSA [Span] gradient decent algorithm orNelder-Mead [NM] algorithm. This would constitute a direct approach tominimizing the energy of |ψ({right arrow over (θ)},{right arrow over(φ)})

with respect to the Hamiltonian H. This corresponds to classicaloptimization routine with the cost function E({right arrow over(θ)},{right arrow over (φ)}) that will be evaluated on a quantumcomputer directly. This approach estimates the optimal energyapproximation E({right arrow over (θ)},{right arrow over (φ)}) ofE_(min) and provides the control parameters to prepare the state|({right arrow over (θ)}*,{right arrow over (φ)}*)

, which is a best estimate for the true state |ψ_(min)

.

A direct minimization approach follows the steps:

-   -   1. For t=0 provide initial parameters ({right arrow over        (θ)}⁰,{right arrow over (φ)}⁰)    -   2. Repeat the following steps M-times:        -   (a) Prepare the state |ψ({right arrow over (θ)}^(t),{right            arrow over (φ)}^(t))            on the quantum computer, and measure a Pauli σ(α) on this            state.        -   (b) Then add the samples to obtain the expectation value            E({right arrow over (θ)}^(t),{right arrow over (φ)}^(t)).    -   3. Then set t→t+1 propose a new set of parameters ({right arrow        over (θ)}^(t+1),{right arrow over (φ)}^(t+1)) based on a        classical optimization scheme    -   4. Repeat until converged to the optimal values ({right arrow        over (θ)}*,{right arrow over (φ)}*).

However, there may be scenarios where it can be favorable to consider aless direct route. A different approach, for instance, is thepossibility of following the adiabatic transformation of a virtualHamiltonian. Note that the classical optimization problem in minimizingE({right arrow over (θ)},{right arrow over (φ)}) directly is most likelynot simpler than minimizing the initial cost function directly. In thiscontext described is a deformation of the cost function depending on aparameter s∈[0,1]. That is, a family of Hamiltonians H(s) is consideredthat interpolate between the initial Hamiltonian H(0)=H_(trivial) to thefinal Hamiltonian H(1)=H_(C). A possible deformation can for example begiven in terms of

H(s)=(1−s)H _(trivial) sH _(C).  (17)

Note that other choices for H(s) are possible as long as the conditionsfor H(0), H(1) are met. The choice of H_(trivial) is made so thatoptimal set of parameters({right arrow over (θ)}*,{right arrow over(φ)}*)|_(s=0) for the ground state of the initial state is known, sothat the ground state |G₀

=|ψ(({right arrow over (θ)}*,{right arrow over (φ)}*))

|_(s=0) with H_(trivial)|G₀

=E_(min) (0)|G₀

can be constructed easily.

To provide a concrete example for non-limiting illustration purposes,consider the MaxCut problem for a graph G=(V,E) with vertex set V andedge set G. The problem can be mapped to minimizing the energy of anIsing Hamiltonian H_(C) as in Eq. (7). As the trivial initialHamiltonian choose H_(trivial)=−Σ_(iεV)X_(i), i.e., the transverse fieldmodel on the graph G. A simple interpolating family is then given by

$\begin{matrix}{{H_{MaxCut}(s)} = {{{- ( {1 - s} )}{\sum\limits_{i \in V}X_{i}}} + {s{\sum\limits_{{({i,k})} \in E}{J_{i,k}Z_{i}{Z_{k}.}}}}}} & (18)\end{matrix}$

The initial state can be |G₀

=|+

^(⊗n). This choice can be readily realized by setting the interactionHamiltonian parameters {right arrow over (φ)} in the trial state of Eq.(8) so that all K=0, and the single qubit rotation parameters {rightarrow over (θ)} so that the first layer of Unitaries is given by H^(⊗n).

The Hamiltonian family H(s) for s∈[0,1] leads to a family of classicalcost functions E(s) that are evaluated on the quantum computer bymeasurement as described above with reference to FIGS. 4 and 5. The costfunction

E _(s)({right arrow over (θ)},{right arrow over (φ)})=

ψ({right arrow over (θ)},{right arrow over (φ)})|H(s)|ψ(({right arrowover (θ)},{right arrow over (φ)})

  (19)

interpolates between the ground state energy approximations E₀,E₁ andupdates parameters ({right arrow over (θ)}*,{right arrow over(φ)}*)|_(s) along the deformed path minimizing the parameters at everystep s using the classical minimizers described with reference to FIG.7.

The method starts with the initial parameters ({right arrow over(θ)},{right arrow over (φ)})|_(s=0), and then follows a schedule for Tvalues 0=s₀<s₁<s₂< . . . <s_(T)=1 of the parameter s, using the optimalvalue ({right arrow over (θ)}*,{right arrow over (φ)}*)|_(s) _(i) as thestarting point for the next minimization. That is

-   -   1. Choose T values 0=s₀<s₁<s₂< . . . <s_(T)=1, and prepare the        initial optimal parameters for s₀=0 as ({right arrow over        (θ)}*,{right arrow over (φ)}*)|_(s=0)    -   2. For 1≤t≤T repeat the following        -   (a) Provide the Hamiltonian H(s_(t))        -   (b) Set the initial parameters ({right arrow over            (θ)},{right arrow over (φ)})=({right arrow over (θ)}*,{right            arrow over (φ)}*)|_(s) _(t=1)        -   (c) Run the optimization method (FIG. 1) for H(s_(t)) and            |φ({right arrow over (θ)},{right arrow over (φ)})    -   3. Report the optimal parameters ({right arrow over (θ)}*,{right        arrow over (φ)}*)|_(s) _(T) and energy E_(s) _(T) .    -   4. Prepare the state |φ({right arrow over (θ)}*_(s) _(T) ,{right        arrow over (φ)}*_(s) _(T) )        and sample multiple times in the computational basis to obtain a        good bit-string that provides an energy of H_(C) comparable to        E_(s) _(T) .

Introducing such a deformation is expected to improve the convergence ofthe classical optimization method, because of performing a warm start inthe parameters ({right arrow over (θ)},{right arrow over (φ)}). Considerthe family of Hamiltonians H_(MaxCut)(s) in Eq. (18), for s∈[0,1]. Notethat other solutions propose a fully classical method to emulate noisyquantum annealing. This classical model can be obtained, within theframework described herein, by considering the very restricted class ofproduct trial states:

$\begin{matrix} {{{{{{\psi ( {\overset{arrow}{\theta},\overset{arrow}{\phi}} )}}\rangle}^{1} = {{\underset{i = 1}{\overset{n}{\otimes}}e^{i\frac{\theta_{i}}{2}Y_{i}}}0^{n}}}\rangle} = {{\underset{i = 1}{\overset{n}{\otimes}}( {{\sin ( \frac{\theta_{i}}{2} )}0}\rangle } - {{\cos ( \frac{\theta_{i}}{2} )}{1\rangle}}}} ) & (20)\end{matrix}$

The technology described herein provides that E_(s)({right arrow over(θ)},{right arrow over (φ)}) can be computed classically, and reduces tothe cost function

$\begin{matrix}{{E_{s}( {\overset{arrow}{\theta},\overset{arrow}{\phi}} )} = {{{- ( {1 - s} )}{\sum\limits_{i = 1}^{n}\; {\sin ( \theta_{i} )}}} + {s{\sum\limits_{i,k}^{n}{J_{i}\mspace{14mu} {\cos ( \theta_{i} )}\mspace{14mu} {{\cos ( \theta_{k} )}.}}}}}} & (21)\end{matrix}$

This cost function corresponds to known classical models. In contrast,the technology described herein corresponds to considering variationalquantum circuits as trial wave functions |ψ({right arrow over(θ)},{right arrow over (φ)})

that have higher depth and actually use entanglement.

As can be seen, described herein is a concrete instantiation of shortdepth circuits for combinatorial optimization problems. The technologymay be mapped to a classical optimization problem for continuousparameters, with results obtained via a deformation from a knownsolution in that space. Cross resonance gates generate that state; thisbasically tunes the state so that it is within the correct class ofstates before sampling. Thus, by minimizing a classical cost functionand applying the circuit to generate output by measuring an energyfunction (which is the cost function on that circuit), the technologyattempts to minimize parameters on the circuit, via a classicalminimization problem. Once known, samples are drawn, which areapproximate solutions to the combinatorial optimization problem tosolve.

FIG. 12 is a representation of a system 1200 that facilitates solving anoptimization problem, e.g., via a memory that stores computer executablecomponents and a processor that executes computer executable componentsstored in the memory. Block 1202 represents a mapping component thatmaps a cost functions to a Hamiltonian based on one or more constraintsto map the optimization problem into an optimization problem overallowed quantum states (e.g., by mapping component 104). Block 1204represents a trial state and measurement component that generates trialstates corresponding to the Hamiltonian by physical time evolution ofquantum hardware interspersed with control pulses to entangle qubits ofthe quantum hardware, and that measures a quantum cost function for thetrial states to determine a trial state that results in optimal values(e.g., by trial state and measurement component 107). Block 1206represents a deformation component that deforms a Hamiltonian into adeformed Hamiltonian to find an optimal state, and uses the optimalstate as a next starting state for a next optimization on the deformedHamiltonian until an optimizer is determined with respect to a desiredHamiltonian (e.g., by deformation component 118).

Aspects can comprise a sampling component that samples from the optimalstate corresponding to the optimizer to obtain one or moreapproximations to the combinatorial optimization problem. Other aspectscan comprise an interface that interacts with the mapping component andoutputs the trial state that results in the optimal values. The trialstate and measurement component can measure individual Pauli operatorson the trial states.

FIG. 13 is a flow diagram representation of example operations generallydirected towards facilitating solving a combinatorial optimizationproblem using quantum hardware. Aspects comprise mapping (block 1302) acost function associated with the combinatorial optimization problem toan optimization problem over allowed quantum states, comprisingconstructing a quantum Hamiltonian for the cost function. Other aspectscomprise generating (block 1304) a set of trial states by a physicaltime evolution of the quantum hardware interspersed with control pulses,measuring (block 1306) a quantum cost function for the trial states,determining (block 1308) a trial state resulting in optimal values, anddeforming (block 1310) a Hamiltonian to find an optimal state and usingthe optimal state as a next starting state for a next optimization on adeformed Hamiltonian until an optimizer is determined with respect to adesired Hamiltonian.

Aspects can comprise generating the set of trial states by the physicaltime evolution of quantum hardware interspersed with control pulsescomprises using the control pulses to generate entanglement.

Constructing the quantum Hamiltonian for the cost function can compriseconstructing a quantum Hamiltonian diagonal. Constructing the quantumHamiltonian for the cost function can comprise expressing a quantumHamiltonian as a linear combination of Pauli Z-terms.

Mapping the cost function associated with the binary combinatorialoptimization problem can comprise mapping the cost function to a qubitrepresentation on the approximate quantum computer. Measuring thequantum cost function for the trial states can comprise measuringindividual Pauli operators on the trial states.

Aspects can comprise sampling from the optimal state corresponding tothe optimizer to obtain approximations to the combinatorial optimizationproblem. The combinatorial optimization problem can comprise a binarycombinatorial optimization problem, and aspects can comprise samplingfrom the optimal state corresponding to the optimizer to obtain bitstrings that provide approximations to the binary optimization problem.

Other aspects, such as implemented via a computer program product, canbe directed towards facilitating solving a combinatorial optimizationproblem. The computer program product can comprise a computer readablestorage medium having program instructions embodied therewith, theprogram instructions executable by a processor to cause the processor tomap a cost function associated with the combinatorial optimizationproblem to an optimization problem over allowed quantum states,comprising constructing a quantum Hamiltonian for the cost function. Thecomputer program product can generate a set of trial states by aphysical time evolution of the quantum hardware interspersed withcontrol pulses, measure a quantum cost function for the trial states,determine a trial state resulting in optimal values, and deform aHamiltonian to find an optimal state and using the optimal state as anext starting state for a next optimization on a deformed Hamiltonianuntil an optimizer is determined with respect to a desired Hamiltonian.

The instructions to generate the set of trial states by the physicaltime evolution of quantum hardware interspersed with control pulses cancomprise instructions to use the control pulses to generateentanglement. Constructing the quantum Hamiltonian for the cost functioncan comprise constructing a quantum Hamiltonian diagonal. Constructingthe quantum Hamiltonian diagonal for the cost function can compriseexpressing a quantum Hamiltonian as a linear combination of PauliZ-terms.

The instructions to map the cost functions associated with the binarycombinatorial optimization problem can comprise instructions to map thecost functions to a qubit representation on the approximate quantumcomputer. The instructions to measure the quantum cost functions for thetrial states can comprise instructions to measure individual Paulioperators on the trial states. The instructions can compriseinstructions to sample via the approximate quantum computer from theoptimal state corresponding to the optimizer to obtain bit strings thatprovide approximations to the optimization problem.

Other aspects, exemplified as operations in FIG. 14, can compriseobtaining a starting Hamiltonian having associated starting controlparameters (operation 1402) and using quantum hardware to deform thestarting Hamiltonian into a deformed Hamiltonian associated with optimalcontrol parameters for that deformed Hamiltonian (operation 1404).Aspects can comprise using the quantum hardware to repeatedly deform thedeformed Hamiltonian with the associated optimal control parameters forthat deformed Hamiltonian into further deformed Hamiltonians and furtheroptimal control parameters associated therewith until a desiredHamiltonian is reached (operation 1406) and outputting informationcorresponding to the desired Hamiltonian and the optimal controlparameters associated with the desired Hamiltonian (operation 1408).

Aspects can comprise sampling based on the information corresponding tothe desired Hamiltonian and the optimal control parameters associatedwith the desired Hamiltonian to obtain data that provide approximationsto a combinatorial optimization problem. Other aspects can comprisegenerating a set of trial states by a physical time evolution of thequantum hardware interspersed with control pulses that generateentanglement, measuring a quantum cost function for the trial states anddetermining a trial state resulting in optimal values.

A computer program product facilitating solving a binary combinatorialoptimization problem can be provided, and the computer program productcan comprise a computer readable storage medium having programinstructions embodied therewith. The program instructions can beexecutable by a processor to cause the processor to obtain a startingHamiltonian having associated starting control parameters, and usequantum hardware to deform the starting Hamiltonian into a deformedHamiltonian associated with optimal control parameters for that deformedHamiltonian. Other instructions can use the quantum hardware torepeatedly deform the deformed Hamiltonian with the associated optimalcontrol parameters for that deformed Hamiltonian into further deformedHamiltonians and further optimal control parameters associated therewithuntil a desired Hamiltonian is reached and output informationcorresponding to the desired Hamiltonian and the optimal controlparameters associated with the desired Hamiltonian.

Other aspects can comprise instructions to sample based on theinformation corresponding to the desired Hamiltonian and the optimalcontrol parameters associated with the desired Hamiltonian to obtaindata that provide approximations to a combinatorial optimizationproblem. Still other aspects can comprise instructions to generate a setof trial states by a physical time evolution of the quantum hardwareinterspersed with control pulses that generate entanglement, measuring aquantum cost function for the trial states and determining a trial stateresulting in optimal values.

In order to provide a context for the various aspects of the disclosedsubject matter, FIG. 15 as well as the following discussion are intendedto provide a general description of a suitable environment in which thevarious aspects of the disclosed subject matter can be implemented. FIG.15 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can befacilitated. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity.

With reference to FIG. 15, a suitable operating environment 1500 forimplementing various aspects of this disclosure can also include acomputer 1512. The computer 1512 can also include a processing unit1514, a system memory 1516, and a system bus 1518. The system bus 1518couples system components including, but not limited to, the systemmemory 1516 to the processing unit 1514. The processing unit 1514 can beany of various available processors. Dual microprocessors and othermultiprocessor architectures also can be employed as the processing unit1514. The system bus 1518 can be any of several types of busstructure(s) including the memory bus or memory controller, a peripheralbus or external bus, and/or a local bus using any variety of availablebus architectures including, but not limited to, Industrial StandardArchitecture (ISA), Micro-Channel Architecture (MSA), Extended ISA(EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB),Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus(USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and SmallComputer Systems Interface (SCSI).

The system memory 1516 can also include volatile memory 1520 andnonvolatile memory 1522. The basic input/output system (BIOS),containing the basic routines to transfer information between elementswithin the computer 1512, such as during start-up, is stored innonvolatile memory 1522. Computer 1512 can also includeremovable/non-removable, volatile/non-volatile computer storage media.FIG. 15 illustrates, for example, a disk storage 1524. Disk storage 1524can also include, but is not limited to, devices like a magnetic diskdrive, floppy disk drive, tape drive, Jaz drive, Zip drive, LS-100drive, flash memory card, or memory stick. The disk storage 1524 alsocan include storage media separately or in combination with otherstorage media. To facilitate connection of the disk storage 1524 to thesystem bus 1518, a removable or non-removable interface is typicallyused, such as interface 1526. FIG. 15 also depicts software that acts asan intermediary between users and the basic computer resources describedin the suitable operating environment 1500. Such software can alsoinclude, for example, an operating system 1528. Operating system 1528,which can be stored on disk storage 1524, acts to control and allocateresources of the computer 1512.

System applications 1530 take advantage of the management of resourcesby operating system 1528 through program modules 1532 and program data1534, e.g., stored either in system memory 1516 or on disk storage 1524.It is to be appreciated that this disclosure can be implemented withvarious operating systems or combinations of operating systems. A userenters commands or information into the computer 1512 through inputdevice(s) 1536. Input devices 1536 include, but are not limited to, apointing device such as a mouse, trackball, stylus, touch pad, keyboard,microphone, joystick, game pad, satellite dish, scanner, TV tuner card,digital camera, digital video camera, web camera, and the like. Theseand other input devices connect to the processing unit 1514 through thesystem bus 1518 via interface port(s) 1538. Interface port(s) 1538include, for example, a serial port, a parallel port, a game port, and auniversal serial bus (USB). Output device(s) 1540 use some of the sametype of ports as input device(s) 1536. Thus, for example, a USB port canbe used to provide input to computer 1512, and to output informationfrom computer 1512 to an output device 1540. Output adapter 1542 isprovided to illustrate that there are some output devices 1540 likemonitors, speakers, and printers, among other output devices 1540, whichrequire special adapters. The output adapters 1542 include, by way ofillustration and not limitation, video and sound cards that provide ameans of connection between the output device 1540 and the system bus1518. It should be noted that other devices and/or systems of devicesprovide both input and output capabilities such as remote computer(s)1544.

Computer 1512 can operate in a networked environment using logicalconnections to one or more remote computers, such as remote computer(s)1544. The remote computer(s) 1544 can be a computer, a server, a router,a network PC, a workstation, a microprocessor based appliance, a peerdevice or other common network node and the like, and typically can alsoinclude many or all of the elements described relative to computer 1512.For purposes of brevity, only a memory storage device 1546 isillustrated with remote computer(s) 1544. Remote computer(s) 1544 islogically connected to computer 1512 through a network interface 1548and then physically connected via communication connection 1550. Networkinterface 1548 encompasses wire and/or wireless communication networkssuch as local-area networks (LAN), wide-area networks (WAN), cellularnetworks, etc. LAN technologies include Fiber Distributed Data Interface(FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ringand the like. WAN technologies include, but are not limited to,point-to-point links, circuit switching networks like IntegratedServices Digital Networks (ISDN) and variations thereon, packetswitching networks, and Digital Subscriber Lines (DSL). Communicationconnection(s) 1550 refers to the hardware/software employed to connectthe network interface 1548 to the system bus 1518. While communicationconnection 1550 is shown for illustrative clarity inside computer 1512,it can also be external to computer 1512. The hardware/software forconnection to the network interface 1548 can also include, for exemplarypurposes only, internal and external technologies such as, modemsincluding regular telephone grade modems, cable modems and DSL modems,ISDN adapters, and Ethernet cards.

The present invention can be a system, a method, an apparatus and/or acomputer program product at any possible technical detail level ofintegration. The computer program product can include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention. The computer readable storage medium can be atangible device that can retain and store instructions for use by aninstruction execution device. The computer readable storage medium canbe, for example, but is not limited to, an electronic storage device, amagnetic storage device, an optical storage device, an electromagneticstorage device, a semiconductor storage device, or any suitablecombination of the foregoing. A non-exhaustive list of more specificexamples of the computer readable storage medium can also include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a static random access memory(SRAM), a portable compact disc read-only memory (CD-ROM), a digitalversatile disk (DVD), a memory stick, a floppy disk, a mechanicallyencoded device such as punch-cards or raised structures in a groovehaving instructions recorded thereon, and any suitable combination ofthe foregoing. A computer readable storage medium, as used herein, isnot to be construed as being transitory signals per se, such as radiowaves or other freely propagating electromagnetic waves, electromagneticwaves propagating through a waveguide or other transmission media (e.g.,light pulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network can comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device. Computer readable programinstructions for carrying out operations of the present invention can beassembler instructions, instruction-set-architecture (ISA) instructions,machine instructions, machine dependent instructions, microcode,firmware instructions, state-setting data, configuration data forintegrated circuitry, or either source code or object code written inany combination of one or more programming languages, including anobject oriented programming language such as Smalltalk, C++, or thelike, and procedural programming languages, such as the “C” programminglanguage or similar programming languages. The computer readable programinstructions can execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer can beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection can be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) can execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions. These computer readable programinstructions can be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks. These computer readable program instructions can also be storedin a computer readable storage medium that can direct a computer, aprogrammable data processing apparatus, and/or other devices to functionin a particular manner, such that the computer readable storage mediumhaving instructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks. Thecomputer readable program instructions can also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational acts to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks can occur out of theorder noted in the Figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the subject matter has been described above in the general contextof computer-executable instructions of a computer program product thatruns on a computer and/or computers, those skilled in the art willrecognize that this disclosure also can or can be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, etc. thatperform particular tasks and/or implement particular abstract datatypes. Moreover, those skilled in the art will appreciate that theinventive computer-implemented methods can be practiced with othercomputer system configurations, including single-processor ormultiprocessor computer systems, mini-computing devices, mainframecomputers, as well as computers, hand-held computing devices (e.g., PDA,phone), microprocessor-based or programmable consumer or industrialelectronics, and the like. The illustrated aspects can also be practicedin distributed computing environments in which tasks are performed byremote processing devices that are linked through a communicationsnetwork. However, some, if not all aspects of this disclosure can bepracticed on stand-alone computers. In a distributed computingenvironment, program modules can be located in both local and remotememory storage devices.

As used in this application, the terms “component,” “system,”“platform,” “interface,” and the like, can refer to and/or can include acomputer-related entity or an entity related to an operational machinewith one or more specific functionalities. The entities disclosed hereincan be either hardware, a combination of hardware and software,software, or software in execution. For example, a component can be, butis not limited to being, a process running on a processor, a processor,an object, an executable, a thread of execution, a program, and/or acomputer. By way of illustration, both an application running on aserver and the server can be a component. One or more components canreside within a process and/or thread of execution and a component canbe localized on one computer and/or distributed between two or morecomputers. In another example, respective components can execute fromvarious computer readable media having various data structures storedthereon. The components can communicate via local and/or remoteprocesses such as in accordance with a signal having one or more datapackets (e.g., data from one component interacting with anothercomponent in a local system, distributed system, and/or across a networksuch as the Internet with other systems via the signal). As anotherexample, a component can be an apparatus with specific functionalityprovided by mechanical parts operated by electric or electroniccircuitry, which is operated by a software or firmware applicationexecuted by a processor. In such a case, the processor can be internalor external to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts, wherein the electroniccomponents can include a processor or other means to execute software orfirmware that confers at least in part the functionality of theelectronic components. In an aspect, a component can emulate anelectronic component via a virtual machine, e.g., within a cloudcomputing system.

In addition, the term “or” is intended to mean an inclusive “or” ratherthan an exclusive “or.” That is, unless specified otherwise, or clearfrom context, “X employs A or B” is intended to mean any of the naturalinclusive permutations. That is, if X employs A; X employs B; or Xemploys both A and B, then “X employs A or B” is satisfied under any ofthe foregoing instances. Moreover, articles “a” and “an” as used in thesubject specification and annexed drawings should generally be construedto mean “one or more” unless specified otherwise or clear from contextto be directed to a singular form. As used herein, the terms “example”and/or “exemplary” are utilized to mean serving as an example, instance,or illustration. For the avoidance of doubt, the subject matterdisclosed herein is not limited by such examples. In addition, anyaspect or design described herein as an “example” and/or “exemplary” isnot necessarily to be construed as preferred or advantageous over otheraspects or designs, nor is it meant to preclude equivalent exemplarystructures and techniques known to those of ordinary skill in the art.

As it is employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. Further, processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of user equipment. A processor can also beimplemented as a combination of computing processing units. In thisdisclosure, terms such as “store,” “storage,” “data store,” datastorage,” “database,” and substantially any other information storagecomponent relevant to operation and functionality of a component areutilized to refer to “memory components,” entities embodied in a“memory,” or components comprising a memory. It is to be appreciatedthat memory and/or memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory. By way of illustration, and not limitation,nonvolatile memory can include read only memory (ROM), programmable ROM(PROM), electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g.,ferroelectric RAM (FeRAM). Volatile memory can include RAM, which canact as external cache memory, for example. By way of illustration andnot limitation, RAM is available in many forms such as synchronous RAM(SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rateSDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM),direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), andRambus dynamic RAM (RDRAM). Additionally, the disclosed memorycomponents of systems or computer-implemented methods herein areintended to include, without being limited to including, these and anyother suitable types of memory.

What has been described above include mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describeevery conceivable combination of components or computer-implementedmethods for purposes of describing this disclosure, but one of ordinaryskill in the art can recognize that many further combinations andpermutations of this disclosure are possible. Furthermore, to the extentthat the terms “includes,” “has,” “possesses,” and the like are used inthe detailed description, claims, appendices and drawings such terms areintended to be inclusive in a manner similar to the term “comprising” as“comprising” is interpreted when employed as a transitional word in aclaim.

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

What is claimed is:
 1. A system, comprising: a memory that storescomputer executable components; a processor that executes computerexecutable components stored in the memory, wherein the computerexecutable components comprise: a mapping component that maps a costfunction to a Hamiltonian based on one or more constraints to map theoptimization problem into an optimization problem over allowed quantumstates; a trial state and measurement component that generates trialstates corresponding to the Hamiltonian by physical time evolution ofquantum hardware interspersed with control pulses to entangle qubits ofthe quantum hardware, and that measures a quantum cost function for thetrial states to determine a trial state that results in optimal values;and a deformation component that deforms a Hamiltonian into a deformedHamiltonian to find an optimal state, and uses the optimal state as anext starting state for a next optimization on the deformed Hamiltonianuntil an optimizer is determined with respect to a desired Hamiltonian.2. The system of claim 1, further comprising a sampling component thatsamples from the optimal state corresponding to the optimizer to obtainone or more approximations to the combinatorial optimization problem. 3.The system of claim 1, further comprising an interface that interactswith the mapping component and outputs the trial state that results inthe optimal values.
 4. The system of claim 1, wherein the trial stateand measurement component measures individual Pauli operators on thetrial states.
 5. A computer-implemented method, comprising:facilitating, by a system operatively coupled to a processor, solving acombinatorial optimization problem using quantum hardware, the systemcomprising: mapping, by the system, a cost function associated with thecombinatorial optimization problem to an optimization problem overallowed quantum states, comprising constructing a quantum Hamiltonianfor the cost function; generating, by the system, a set of trial statesby a physical time evolution of the quantum hardware interspersed withcontrol pulses; measuring, by the system, a quantum cost function forthe trial states; determining, by the system, a trial state resulting inoptimal values; and deforming, by the system, a Hamiltonian to find anoptimal state and using the optimal state as a next starting state for anext optimization on a deformed Hamiltonian until an optimizer isdetermined with respect to a desired Hamiltonian.
 6. Thecomputer-implemented method of claim 5, wherein the generating the setof trial states by the physical time evolution of quantum hardwareinterspersed with control pulses comprises using the physical timeevolution to generate entanglement.
 7. The computer-implemented methodof claim 5, wherein the constructing the quantum Hamiltonian for thecost function comprises constructing a quantum Hamiltonian diagonal. 8.The computer-implemented method of claim 5, wherein the constructing thequantum Hamiltonian for the cost function comprises expressing a quantumHamiltonian as a linear combination of Pauli Z-terms.
 9. Thecomputer-implemented method of claim 5, wherein the mapping the costfunction associated with the binary combinatorial optimization problemcomprises mapping the cost function to a qubit representation on theapproximate quantum computer.
 10. The computer-implemented method ofclaim 5, wherein the measuring the quantum cost function for the trialstates comprises measuring individual Pauli operators on the trialstates.
 11. The computer-implemented method of claim 5, furthercomprising, sampling from the optimal state corresponding to theoptimizer to obtain approximations to the combinatorial optimizationproblem.
 12. The computer-implemented method of claim 5, wherein thecombinatorial optimization problem comprises a binary combinatorialoptimization problem, and further comprising sampling from the optimalstate corresponding to the optimizer to obtain bit strings that provideapproximations to the binary optimization problem.
 13. A computerprogram product facilitating solving a combinatorial optimizationproblem, the computer program product comprising a computer readablestorage medium having program instructions embodied therewith, theprogram instructions executable by a processor to cause the processorto: map a cost function associated with the combinatorial optimizationproblem to an optimization problem over allowed quantum states,comprising constructing a quantum Hamiltonian for the cost function;generate a set of trial states by a physical time evolution of thequantum hardware interspersed with control pulses; measure a quantumcost function for the trial states; determine a trial state resulting inoptimal values; and deform a Hamiltonian to find an optimal state andusing the optimal state as a next starting state for a next optimizationon a deformed Hamiltonian until an optimizer is determined with respectto a desired Hamiltonian.
 14. The computer program product of claim 13,wherein the program instructions are further executable by the processorto cause the processor to generate the set of trial states by thephysical time evolution of quantum hardware interspersed with controlpulses comprises instructions to use the control pulses to generateentanglement.
 15. The computer program product of claim 13, wherein theconstructing the quantum Hamiltonian for the cost function comprisesconstructing a quantum Hamiltonian diagonal.
 16. The computer programproduct of claim 13, wherein the constructing the quantum Hamiltoniandiagonal for the cost function comprises expressing a quantumHamiltonian as a linear combination of Pauli Z-terms.
 17. The computerprogram product of claim 13, wherein the instructions to map the costfunctions associated with the binary combinatorial optimization problemcomprises instructions to map the cost functions to a qubitrepresentation on the approximate quantum computer.
 18. The computerprogram product of claim 13, wherein the instructions to measure thequantum cost functions for the trial states comprises instructions tomeasure individual Pauli operators on the trial states.
 19. The computerprogram product of claim 13, wherein the program instructions arefurther executable by the processor to cause the processor to sample viathe approximate quantum computer from the optimal state corresponding tothe optimizer to obtain bit strings that provide approximations to theoptimization problem.
 20. A computer-implemented method, comprising:obtaining, by a system operatively coupled to a processor, a startingHamiltonian having associated starting control parameters; using, by thesystem, quantum hardware to deform the starting Hamiltonian into adeformed Hamiltonian associated with optimal control parameters for thatdeformed Hamiltonian; using, by the system, the quantum hardware torepeatedly deform the deformed Hamiltonian with the associated optimalcontrol parameters for that deformed Hamiltonian into further deformedHamiltonians and further optimal control parameters associated therewithuntil a desired Hamiltonian is reached; and outputting, by the system,information corresponding to the desired Hamiltonian and the optimalcontrol parameters associated with the desired Hamiltonian.
 21. Thecomputer-implemented method of claim 20, further comprising sampling, bythe system, based on the information corresponding to the desiredHamiltonian and the optimal control parameters associated with thedesired Hamiltonian to obtain data that provide approximations to acombinatorial optimization problem.
 22. The computer-implemented methodof claim 20, further comprising generating, by the system, a set oftrial states by a physical time evolution of the quantum hardwareinterspersed with control pulses that generate entanglement, measuring aquantum cost function for the trial states and determining a trial stateresulting in optimal values.
 23. A computer program product facilitatingsolving a binary combinatorial optimization problem, the computerprogram product comprising a computer readable storage medium havingprogram instructions embodied therewith, the program instructionsexecutable by a processor to cause the processor to: obtain a startingHamiltonian having associated starting control parameters; use quantumhardware to deform the starting Hamiltonian into a deformed Hamiltonianassociated with optimal control parameters for that deformedHamiltonian; use the quantum hardware to repeatedly deform the deformedHamiltonian with the associated optimal control parameters for thatdeformed Hamiltonian into further deformed Hamiltonians and furtheroptimal control parameters associated therewith until a desiredHamiltonian is reached; and output information corresponding to thedesired Hamiltonian and the optimal control parameters associated withthe desired Hamiltonian.
 24. The computer program product of claim 23,wherein the program instructions are further executable by the processorto cause the processor to sample based on the information correspondingto the desired Hamiltonian and the optimal control parameters associatedwith the desired Hamiltonian to obtain data that provide approximationsto a combinatorial optimization problem.
 25. The computer programproduct of claim 23, wherein the program instructions are furtherexecutable by the processor to cause the processor to generate a set oftrial states by a physical time evolution of the quantum hardwareinterspersed with control pulses that generate entanglement, measuring aquantum cost function for the trial states and determining a trial stateresulting in optimal values.